Drive value with no input
I don't think it's possible to directly drive the brick texture Offset from an input socket but you can formulate nodes to adjust the input Vector to achieve the same effect using the following nodes :



Note the two Value nodes on the left hand side - these can be used to adjust the Scale and Offset (the actual Brick Texture's Offset can be left at Zero). Adjusting the Scale or Offset should produce a similar effect to that of the actual Brick texture node.



This works by using the Scale and Y coordinate to calculate (using a Modulo node) where each alternate row of bricks starts and ends, and then calculating the offset (again, based on the Scale) and adding this into the X coordinate to offset those rows of bricks.

Note that this is only currently capable of adjusting the Scale and Offset - additional work will be required on the Maths nodes to allow for adjusting some of the other brick values (such as the Row Height).

Blend file included